On the Local Langlands Functoriality Transfer From $\text{SO}(5)$ to $\text{GL}(4)$

We study the local Langlands functoriality transfer from $\text{SO}(5, F)$ to $\text{GL}(4, F)$ for arbitrary twists of several families of irreducible supercuspidal representations of $\text{GL}(4, F)$, where $F$ is a non-archimedean local field of characteristic zero. In doing so, we give equivalent conditions for such representations to be functoriality transfers from $\text{SO}(5, F)$ in terms of the Bushnell-Kutzko construction of supercuspidal representations by studying poles of local exterior square $L$-functions and the existence of non-zero local Shalika models. This article provides a starting point for an explicit characterization of this functoriality transfer in terms of type theory.